Bulk Universality and Related Properties of Hermitian Matrix Models

We give a new proof of universality properties in the bulk of spectrum of the hermitian matrix models, assuming that the potential that determines the model is globally C ² and locally C ³ function (see Theorem 3.1). The proof as our previous proof in (Pastur and Shcherbina in J. Stat. Phys. 86:109–147, 1997) is based on the orthogonal polynomial techniques but does not use asymptotics of orthogonal polynomials. Rather, we obtain the sin -kernel as a unique solution of a certain non-linear integro-differential equation that follows from the determinant formulas for the correlation functions of the model. We also give a simplified and strengthened version of paper (Boutet de Monvel, et al. in J. Stat. Phys. 79:585–611, 1995) on the existence and properties of the limiting Normalized Counting Measure of eigenvalues. We use these results in the proof of universality and we believe that they are of independent interest.     

On Universality for Orthogonal Ensembles of Random Matrices

We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive Toeplitz matrix and a two diagonal skew symmetric Toeplitz matrix.     

Linear Statistics of Point Processes via Orthogonal Polynomials

For arbitrary β>0, we use the orthogonal polynomials techniques developed in (Killip and Nenciu in , 2005; Killip and Nenciu in Int. Math. Res. Not. 50: 2665–2701, 2004) to study certain linear statistics associated with the circular and Jacobi β ensembles. We identify the distribution of these statistics then prove a joint central limit theorem. In the circular case, similar statements have been proved using different methods by a number of authors. In the Jacobi case these results are new.     

Nonintersecting Brownian Motions on the Half-Line and Discrete Gaussian Orthogonal Polynomials

We study the distribution of the maximal height of the outermost path in the model of N nonintersecting Brownian motions on the half-line as N→∞, showing that it converges in the proper scaling to the Tracy-Widom distribution for the largest eigenvalue of the Gaussian orthogonal ensemble. This is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the Airy₂ process minus a parabola. Our proof is based on Riemann-Hilbert analysis of a system of discrete orthogonal polynomials with a Gaussian weight in the double scaling limit as this system approaches saturation. We consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation. Both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point.     

Universality in Orthogonal and Symplectic Invariant Matrix Models with Quartic Potential

In this work, we develop an orthogonal-polynomials approach for random matrices with orthogonal or symplectic invariant laws, called one-matrix models with polynomial potential in theoretical physics, which are a generalization of Gaussian random matrices. The representation of the correlation functions in these matrix models, via the technique of quaternion determinants, makes use of matrix kernels. We get new formulas for matrix kernels, generalizing the known formulas for Gaussian random matrices, which essentially express them in terms of the reproducing kernel of the theory of orthogonal polynomials. Finally, these formulas allow us to prove the universality of the local statistics of eigenvalues, both in the bulk and at the edge of the spectrum, for matrix models with two-band quartic potential by using the asymptotics given by Bleher and Its for the corresponding orthogonal polynomials.     

Three-Qubit Entangled Embeddings of <Emphasis Type="Italic">CPT</Emphasis> and Dirac Groups within <Emphasis Type="Italic">E</Emphasis><Subscript>8</Subscript> Weyl Group

In quantum information context, the groups generated by Pauli spin matrices, and Dirac gamma matrices, are known as the single qubit Pauli group ℘, and two-qubit Pauli group ℘₂, respectively. It has been found (Socolovsky, Int. J. Theor. Phys. 43: 1941, 2004) that the CPT group of the Dirac equation is isomorphic to ℘. One introduces a two-qubit entangling orthogonal matrix S basically related to the CPT symmetry. With the aid of the two-qubit swap gate, the S matrix allows the generation of the three-qubit real Clifford group and, with the aid of the Toffoli gate, the Weyl group W(E ₈) is generated (Planat, Preprint , 2009). In this paper, one derives three-qubit entangling groups [(P)\tilde]\tilde{\mathcal{P}} and [(P)\tilde]₂\tilde{\mathcal{P}}_{2}, isomorphic to the CPT group ℘ and to the Dirac group ℘₂, that are embedded into W(E ₈). One discovers a new class of pure three-qubit quantum states with no-vanishing concurrence and three-tangle that we name CPT states. States of the GHZ and CPT families, and also chain-type states, encode the new representation of the Dirac group and its CPT subgroup.     

Comment on “Local Copying of d×d-dimensional Partially Entangled Pure States”

Recently, Li et al. (Int. J. Theor. Phys. 48:2777, 2009) derived a necessary and sufficient condition for LOCC cloning of a set of bipartite orthogonal partially but equally entangled state. We demonstrates that, the result is based on a wrong observation regarding a set of non-maximally entangled states with equal entanglement. We also provide a simple example in favor of our comment.     

Rare Event Simulation for T-cell Activation

The problem of statistical recognition is considered, as it arises in immunobiology, namely, the discrimination of foreign antigens against a background of the body’s own molecules. The precise mechanism of this foreign-self-distinction, though one of the major tasks of the immune system, continues to be a fundamental puzzle. Recent progress has been made by van den Berg, Rand, and Burroughs (J. Theor. Biol. 209:465–486, 2001), who modelled the probabilistic nature of the interaction between the relevant cell types, namely, T-cells and antigen-presenting cells (APCs). Here, the stochasticity is due to the random sample of antigens present on the surface of every APC, and to the random receptor type that characterises individual T-cells. It has been shown previously (van den Berg et al. in J. Theor. Biol. 209:465–486, 2001; Zint et al. in J. Math. Biol. 57:841–861, 2008) that this model, though highly idealised, is capable of reproducing important aspects of the recognition phenomenon, and of explaining them on the basis of stochastic rare events. These results were obtained with the help of a refined large deviation theorem and were thus asymptotic in nature. Simulations have, so far, been restricted to the straightforward simple sampling approach, which does not allow for sample sizes large enough to address more detailed questions. Building on the available large deviation results, we develop an importance sampling technique that allows for a convenient exploration of the relevant tail events by means of simulation. With its help, we investigate the mechanism of statistical recognition in some depth. In particular, we illustrate how a foreign antigen can stand out against the self background if it is present in sufficiently many copies, although no a priori difference between self and nonself is built into the model.     

Fluctuations of Matrix Entries of Regular Functions of Wigner Matrices

We study the fluctuations of the matrix entries of regular functions of Wigner random matrices in the limit when the matrix size goes to infinity. In the case of the Gaussian ensembles (GOE and GUE) this problem was considered by A. Lytova and L. Pastur (J. Stat. Phys. 134:147–159, 2009). Our results are valid provided the off-diagonal matrix entries have finite fourth moment, the diagonal matrix entries have finite second moment, and the test functions have four continuous derivatives in a neighborhood of the support of the Wigner semicircle law. Moreover, if the marginal distributions satisfy the Poincaré inequality our results are valid for Lipschitz continuous test functions.     

Combining Rare Events Techniques: Phase Change in Si Nanoparticles

We describe a combined Restrained MD/Parallel Tempering approach to study the difference in free energy as a function of a set of collective variables between two states in presence of metastabilities in the manifold orthogonal to the one spanned by the chosen collective variables. We illustrate the method by an extended study of the relative stability of the amorphous vs crystalline Si nanoparticles embedded in a-SiO₂ of size ranging between 0.8 and 1.8 nm as a function of temperature [S. Orlandini, S. Meloni, and L. Colombo in Phys. Rev. B 83:235303, 2011]. The results show that the method permits to get over the hidden metastabilities. Finally, we try to identify the missing collective variables from the Restrained MD/Parallel Tempering trajectories and analyze whether the collective variable used to control the amorphous-to-crystalline transition is adequate to describe the mechanism of crystallization of some of the nanoparticles considered.     

Black Hole Entropy: From Shannon to Bekenstein

In this note we have applied directly the Shannon formula for information theory entropy to derive the Black Hole (Bekenstein-Hawking) entropy. Our analysis is semi-classical in nature since we use the (recently proposed Banerjee in Int. J. Mod. Phys. D 19:2365–2369, 2010 and Banerjee and Majhi in Phys. Rev. D 81:124006, 2010; Phys. Rev. D 79:064024, 2009; Phys. Lett. B 675:243, 2009) quantum mechanical near horizon mode functions to compute the tunneling probability that goes in to the Shannon formula, following the general idea of Brillouin (Science and Information Theory, Dover, New York, 2004). Our framework conforms to the information theoretic origin of Black Hole entropy, as originally proposed by Bekenstein.     

Quantum-Like Model for Decision Making Process in?Two Players Game A Non-Kolmogorovian Model

In experiments of games, players frequently make choices which are regarded as irrational in game theory. In papers of Khrennikov (Information Dynamics in Cognitive, Psychological and Anomalous Phenomena. Fundamental Theories of Physics, Kluwer Academic, Norwell, 2004; Fuzzy Sets Syst. 155:4–17, 2005; Biosystems 84:225–241, 2006; Found. Phys. 35(10):1655–1693, 2005; in QP-PQ Quantum Probability and White Noise Analysis, vol. XXIV, pp. 105–117, 2009), it was pointed out that statistics collected in such the experiments have “quantum-like” properties, which can not be explained in classical probability theory. In this paper, we design a simple quantum-like model describing a decision-making process in a two-players game and try to explain a mechanism of the irrational behavior of players. Finally we discuss a mathematical frame of non-Kolmogorovian system in terms of liftings (Accardi and Ohya, in Appl. Math. Optim. 39:33–59, 1999).     

Singular Harmonic Maps and Applications to General Relativity

The study of axially symmetric stationary multi-black-hole configurations and the force between co-axially rotating black holes involves, as a first step, an analysis on the “boundary regularity” of the so-called reduced singular harmonic maps. We carry out this analysis by considering those harmonic maps as solutions to some homogeneous divergence systems of partial differential equations with singular coefficients. Our results extend previous works by Weinstein (Comm Pure Appl Math 43:903–948, 1990; Comm Pure Appl Math 45:1183–1203, 1992) and by Li and Tian (Manu Math 73(1):83–89, 1991; Commun Math Phys 149:1–30, 1992; Differential geometry: PDE on manifolds, vol 54, pp. 317–326, 1993). This paper is based on the Ph.D. thesis of the author (Singular harmonic maps into hyperbolic spaces and applications to general relativity, PhD thesis, The State University of New Jersey, Rutgers, 2009).     

Non-uniqueness Results for Critical Metrics of Regularized Determinants in Four Dimensions

The regularized determinant of the Paneitz operator arises in quantum gravity [see Connes in (Noncommutative geometry, 1994), IV.4.γ]. An explicit formula for the relative determinant of two conformally related metrics was computed by Branson in (Commun Math Phys 178:301–309, 1996). A similar formula holds for Cheeger’s half-torsion, which plays a role in self-dual field theory [see Juhl in (Families of conformally covariant differential operators, q-curvature and holography. Progress in Mathematics, vol 275, 2009)], and is defined in terms of regularized determinants of the Hodge laplacian on p-forms (p < n/2). In this article we show that the corresponding actions are unbounded (above and below) on any conformal four-manifold. We also show that the conformal class of the round sphere admits a second solution which is not given by the pull-back of the round metric by a conformal map, thus violating uniqueness up to gauge equivalence. These results differ from the properties of the determinant of the conformal Laplacian established in (Commun Math Phys 149:241–262, 1992), (Ann Math 142:171–212, 1995), (Commun Math Phys 189:655–665, 1997).     

On nonlinear coherent states properties for electron-phonon dynamics

This work addresses a construction of a dual pair of nonlinear coherent states (NCS) in the context of changes of bases in the underlying Hilbert space for a model pertaining to an electron-phonon model in the condensed matter physics, obeying a f-deformed Heisenberg algebra. The existence and properties of reproducing kernel in the NCS Hilbert space are studied and discussed; the probability density and its dynamics in the basis of constructed coherent states are provided. A Glauber-Sudarshan P-representation of the density matrix and relevant issues related to the reproducing kernel properties are presented. Moreover, a NCS quantization of classical phase space observables is performed and illustrated in a concrete example of q-deformed coherent states. Finally, an exposition of quantum optical properties is given.     

<Emphasis Type="Italic">N</Emphasis>=2 de Sitter Super-symmetry Algebra

It was shown that N=1 super-symmetry algebra can be constructed in de Sitter space (Pahlavan et al. in Phys Lett. B 627:217–223, 2005), through calculation of charge conjugation in the ambient space notation (Moradi et al. in Phys. Lett. B 613:74, 2005; Phys. Lett. B 658:284, 2008). Calculation of N=2 super-symmetry algebra constitutes the main frame of this paper. N=2 super-symmetry algebra was presented in Pilch et al. (Commun. Math. Phys. 98:105, 1985). In this paper, we obtain an alternative N=2 super-symmetry algebra.